Computer Vision Group Prof. Daniel Cremers. 4a. Inference in Graphical Models

Transcription

1 Group Prof. Daniel Cremers 4a. Inference in Graphical Models

2 Inference on a Chain (Rep.) The first values of µ α and µ β are: The partition function can be computed at any node: Overall, we have O(NK 2 ) operations to compute the marginal 2 Group

3 More General Graphs The message-passing algorithm can be extended to more general graphs: Undirected Tree Directed Tree Polytree It is then known as the sum-product algorithm. A special case of this is belief propagation. 3 Group

4 More General Graphs The message-passing algorithm can be extended to more general graphs: Undirected Tree An undirected tree is defined as a graph that has exactly one path between any two nodes 4 Group

5 More General Graphs The message-passing algorithm can be extended to more general graphs: A directed tree has only one node without parents and all other nodes have exactly one parent Directed Tree Conversion from a directed to an undirected tree is no problem, because no links are inserted The same is true for the conversion back to a directed tree 5 Group

6 More General Graphs The message-passing algorithm can be extended to more general graphs: Polytrees can contain nodes with several parents, therefore moralization can remove independence relations Polytree 6 Group

7 Factor Graphs The Sum-product algorithm can be used to do inference on undirected and directed graphs. A representation that generalizes directed and undirected models is the factor graph. f(x 1,x 2,x 3 )=p(x 1 )p(x 2 )p(x 3 x 1,x 2 ) Directed graph Factor graph 7 Group

8 Factor Graphs The Sum-product algorithm can be used to do inference on undirected and directed graphs. A representation that generalizes directed and undirected models is the factor graph. Undirected graph Factor graph 8 Group

9 Factor Graphs Factor graphs can contain multiple factors x 1 x 3 f a for the same nodes f b are more general than undirected graphs x 4 are bipartite, i.e. they consist of two kinds of nodes and all edges connect nodes of different kind 9 Group

10 Factor Graphs Directed trees convert to x 1 x 3 tree-structured factor graphs The same holds for x 4 undirected trees Also: directed polytrees convert to tree-structured factor graphs x 1 x 3 f a And: Local cycles in a directed graph can be removed by converting to a x 4 factor graph 10 Group

11 The Sum-Product Algorithm Assumptions: all variables are discrete the factor graph has a tree structure The factor graph represents the joint distribution as a product of factor nodes: p(x) = Y s f s (x s ) The marginal distribution at a given node x is p(x) = X x\x p(x) 11 Group

12 The Sum-Product Algorithm For a given node x the joint can be written as p(x) = Y s2ne(x) F s (x, X s ) Thus, we have p(x) = X x\x Y s2ne(x) F s (x, X s ) Product of all factors associated with f s Key insight: Sum and product can be exchanged! p(x) = Y s2ne(x) X X s F s (x, X s )= Y s2ne(x) µ fs!x(x) Messages from factors to node x 12 Group

13 The Sum-Product Algorithm The factors in the messages can be factorized further: F s (x, X s )=f s (x, x 1,...,x M )G 1 (x 1,X s1 )...G M (x M,X sm ) The messages can then be computed as µ fs!x(x) = X x 1 X = X x 1 X x M f s (x, x 1,...,x M ) x M f s (x, x 1,...,x M ) Y m2ne(f s )\x Y X X s m m2ne(f s )\x G m (x m,x sm ) µ xm!f s (x m ) Messages from nodes to factors 13 Group

14 The Sum-Product Algorithm µ xm!f s (x m )= = Y l2ne(x m )\f s The factors G of the neighboring nodes can again be factorized further: G M (x m,x sm )= X X ml F l (x m,x ml ) Y l2ne(x m )\f s F l (x m,x ml ) This results in the exact same situation as above! We can now recursively apply the derived rules: Y l2ne(x m )\f s µ fl!x m (x m ) 14 Group

15 The Sum-Product Algorithm Summary marginalization: 1.Consider the node x as a root note 2.Initialize the recursion at the leaf nodes as: (var) or µ f!x (x) =1 µ x!f (x) =f(x) (fac) 3.Propagate the messages from the leaves to the root x 4.Propagate the messages back from the root to the leaves 5.We can get the marginals at every node in the graph by multiplying all incoming messages 15 Group

16 The Max-Sum Algorithm Sum-product is used to find the marginal distributions at every node, but: How can we find the setting of all variables that maximizes the joint probability? And what is the value of that maximal probability? Idea: use sum-product to find all marginals and then report the value for each node x that maximizes the marginal p(x) However: this does not give the overall maximum of the joint probability 16 Group

17 The Max-Sum Algorithm Observation: the max-operator is distributive, just like the multiplication used in sum-product: max(ab, ac) =a max(b, c) if a 0 Idea: use max instead of sum as above and exchange it with the product Chain example: max x p(x) = 1 Z max x 1...max[ 1,2 (x 1,x 2 )... N 1,N (x N 1,x N )] = 1 Z max x 1 [ 1,2 (x 1,x 2 )[...max N 1,N (x N 1,x N )]] Message passing can be used as above! 17 Group

18 The Max-Sum Algorithm To find the maximum value of p(x), we start again at the leaves and propagate to the root. Two problems: no summation, but many multiplications; this leads to numerical instability (very small values) when propagating back, multiple configurations of x can maximize p(x), leading to wrong assignments of the overall maximum Solution to the first: Transform everything into log-space and use sums 18 Group

19 The Max-Sum Algorithm Solution to the second problem: Keep track of the arg max in the forward step, i.e. store at each node which value was responsible for the maximum: (x n ) = arg max x n 1 [ln f n 1,n (x n 1,x n )+µ xn 1!f n 1,n (x n )] Then, in the back-tracking step we can recover the arg max by recursive substitution of: x max n 1 = (x max n ) 19 Group

20 Other Inference Algorithms Junction Tree Algorithm: Provides exact inference on general graphs. Works by turning the initial graph into a junction tree and then running a sum-product-like algorithm A junction tree is obtained from an undirected model by triangulation and mapping cliques to nodes and connections of cliques to edges It is the maximal spanning tree of cliques Problem: Intractable on graphs with large cliques. Cost grows exponentially with the number of variables in the largest clique ( tree width ). 20 Group

21 Other Inference Algorithms Loopy Belief Propagation: Performs Sum-Product on general graphs, particularly when they have loops Propagation has to be done several times, until a convergence criterion is met No guarantee of convergence and no global optimum Messages have to be scheduled Initially, unit messages passed across all edges Approximate, but tractable for large graphs 21 Group

22 Conditional Random Fields Another kind of undirected graphical model is known as Conditional Random Field (CRF). CRFs are used for classification where labels are represented as discrete random variables y and features as continuous random variables x A CRF represents the conditional probability where w are parameters learned from training data. CRFs are discriminative and MRFs are generative 22 Group

23 Conditional Random Fields Derivation of the formula for CRFs: In the training phase, we compute parameters w that maximize the posterior: where (x *,y * ) is the training data and p(w) is a Gaussian prior. In the inference phase we maximize 23 Group

24 Conditional Random Fields Typical example: observed variables x i,j are intensity values of pixels in an image and hidden variables y i,j are object labels Note: the definition of x i,j and y i,j is different from the one in C.M. Bishop (pg.389)! 24 Group

25 CRF Training We minimize the negative log-posterior: Computing the likelihood is intractable, as we have to compute the partition function for each w. We can approximate the likelihood using pseudo-likelihood: where Markov blanket C i : All cliques containing y i 25 Group

26 Pseudo Likelihood 26 Group

27 Pseudo Likelihood Pseudo-likelihood is computed only on the Markov blanket of y i and its corresp. feature nodes. 27 Group

28 Potential Functions The only requirement for the potential functions is that they are positive. We achieve that with: where f is a compatibility function that is large if the labels y C fit well to the features x C. This is called the log-linear model. The function f can be, e.g. a local classifier 28 Group

29 Training: CRF Training and Inference Using pseudo-likelihood, training is efficient. We have to minimize: Log-pseudo-likelihood Gaussian prior This is a convex function that can be minimized using gradient descent Inference: Only approximatively, e.g. using loopy belief propagation 29 Group

30 Summary Undirected Graphical Models represent conditional independence more intuitively using graph separation Their factorization is done based on potential functions The normalizer is called the partition function, which in general is intractable to compute Inference in graphical models can be done efficiently using the sum-product algorithm (message passing). Another inference algorithm is loopy belief propagation, which is approximate, but tractable Conditional Random Fields are a special kind of MRFs and can be used for classification 30 Group